Acerbi has several backtests for expected shortfall. The second backtest is based on this equality enter image description hereenter image description here

Does anybody know how to derive this equality? Can anybody explain, why it makes sense, especially dividing by $\alpha$?

Background: I took the equality from this presentation https://www.cass.city.ac.uk/faculty-and-research/faculties/finance/seminars-and-workshops/financial-engineering-workshops/ACERBI-Carlo-10.03.2015.pdf

This is how I know expected shortfall

enter image description here


1 Answer 1


Letting $X_t$ be a random variable, its conditional expectation with respect to some event $E$ is given by:

$$ \mathbb{E}^{\mathbb{P}}[X_t|E]=\frac{\mathbb{E}^{\mathbb{P}}[\mathbf{1}_EX_t]}{\mathbb{P}(E)}$$

In our case, Expected Shortfall is defined as:

$$ \text{ES}_{\alpha,t} = -\mathbb{E}^{\mathbb{P}}[X_t|X_t + \text{VaR}_{\alpha} < 0]$$


$$ \begin{align} \text{ES}_{\alpha,t} & = -\frac{\mathbb{E}^{\mathbb{P}}[\mathbf{1}_{\{X_t + \text{VaR}_{\alpha} < 0\}}X_t]}{\mathbb{P}(X_t + \text{VaR}_{\alpha} < 0)} \\[9pt] & = -\frac{\mathbb{E}^{\mathbb{P}}[\mathbf{1}_{\{X_t + \text{VaR}_{\alpha} < 0\}}X_t]}{\alpha} \\[11pt] & = -\mathbb{E}^{\mathbb{P}}\left[\frac{\mathbf{1}_{\{X_t + \text{VaR}_{\alpha} < 0\}}X_t}{\alpha}\right] \end{align} $$

Where the second step is a consequence of the definition of $\text{VaR}_{\alpha}.$


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